ESPCI-MICHELIN Workshop, December, 2010
Theories of Activated Slow Relaxation, Aging and Mechanical Response of Glassy Polymer & Colloidal Materials
Ken Schweizer
University of Illinois @ Urbana-Champaign, USA
GOALS: “microscopic” statistical dynamical theory
unified approach: highly viscous liquids, glasses, “gels”
quiescent relaxation, aging, nonlinear viscoelasticity
connection to material-specific forces and structure
COWORKERS: Erica Saltzman (liquids) Dr. Kang Chen (glasses)
NSF-NIRT on glassy polymers with Mark Ediger, Jim Caruthers, Juan dePablo
Polymer Reviews: J.Phys.-Condensed Matter, 2009; Annual Rev.Condensed Matter Phys, 2010
Glassy Particle Suspensions
S(q)
q
r/D
0.0 0.5 1.0 1.5 2.0 2.5 3.0
g(r)
0
2
4
6
8
10
φ = 0.6
0.53
0.45g(r)
∗
∗
“Diverging” Relaxation Time, τα ?
r/σ
*
“cage”
* Tunable & Diverse Forces
* Nonspherical Colloid Shapes
* Soft & Deformable
Strong NONgaussian “Dynamic Heterogeneity” Effects
“Trapped” in a “dynamical precursor” world
BUT, still see Activated Hopping, rare event transport
Brownian time : τ0 =σ2 /DSE ~ 0.01-30 secparticle diameter ~ 100 nm -2 µm
WeeksWeitz
Hard Spheres
CAGE
intermittent trajectories
“kinetic vitrify” : τα ∼ 100 − 10,000 sec
STRUCTURE
Nonlinear Langevin Eqn Theory
!̂s(!r,t)="
!r#!ri(t)$
%&'
DERIVATION : KSS, JCP, 2005
Solid State View
r(t) = scalar displacement of a spherical particle
Saltzman & KSS JCP, 2003
Physical Ideas & Technical Approximations
* Key “slow variable” : Density Fluctuations
* Average over local packings: dynamical caging constraints via Effective forces S(q), g(r)
** Local Equilibrium Approx: relate 1 and 2 body dynamics
!"̂s(!r,t)
!t= D
s#2"̂
s(!r,t) + D
s#"̂
s(!r,t) d
!r '$ "̂(!r ',t)#V(
!r %!r ') + &
i#"̂
s(r,t)
!(2)!r,!r ';t( )
!(1)!r;t( )
" !g |!r#!r '|( )
Dynamic “closure”
Ds : dissipative, short time, “bare” processFormally:
ala DDFT
……a theory for single particle trajectories…seek Stochastic Eqn-of -Motion
!s"r(t)"t
= # ""rFeff(r(t))+ $(t)
white noise
!Feff(r) ="3ln(r)"1
3
d!q
(2# )3C2(q)$S(q)e
"q2r2 1+S"1(q)( )/6% & F
ideal+ F
cage
FAVORS: Delocalized Localized Liquid Solid
Nonlinear Langevin Eqn Theory …force balance in overdamped regime
Mean Square Caging Force from Structure
Time Local Displacement-Dependent “Field”
compete
Instantaneous Force on MOVING Particle due to Surrounding particles
“Dynamic Free Energy” =
FULL Dynamics ~ Sequence of locally complex in space-time noise-driven stochastic“events”
!!C(r)
!!C(r)
ρS
Dynamic Free Energy : Hard Spheres
φ = 0.3, 0.46,0.53,0.57,0.6
Displacement, r(t)
..
. .
*
*
rLOC
*
*
r(t)
“MCT transition” = Dynamic Crossover
Transient Localization & Hopping
φC ~ 0.432
Activated Transport
Hard Spheres
nMCT Freezing Kinetic Vitrify
φ
∗
Entropic Barrier
FB(φ)
~7-8 kBT
*“solid” glass
“normal” regime
Analytic Analysis
FB
∗∗
∗
rL
rBrrxn∗ Kramers theory: mean first passage time
!hop!0
=2" (#s /#0
)
K0KB
eFB ~ alpha time
@ cage scale
High Barrier Limit : Real Space Picture
!g2(" ) # F
B “contacts”
FB !"g2(# ) ! "
RCP$"( )
$2%&
“SOLID” only at RCP Jamming
Double Pole NOTWLF free volume
Impulsive Collisions
“mean square force” on moving particle
!s"r(t)"t
= # ""rFeff(r(t))+ $(t)
Noise-Driven Intermittency Trajectory Fluctuations
ALL single particle time correlations…. Heterogeneous Dynamics
Full Numerical Analysis
Alpha (cage scale) Relaxation
! * /!0 " exp F
B(#)( )
!
0.40 0.44 0.48 0.52 0.56
"*
0.1
1
10
100
q*
σ = 1 µm
τ α φ
5x104 s 0.57(14 hrs)
5 months 0.61 "glass"
Fs(q*,t)
approachRCP
! exp B / ("RCP
#")2( )2007 NLEprediction
Cipelletti et al,PRL, 2009extra 2 orders magnitude
MCT
~ double essential singularity
~ new Expts
!"
!0
φC,theory
MANY Heterogeneous Dynamics consequences
Diverse Singular forms *FIT* over ~ first 3 orders magnitude MCT critical power law, “free volume”, ……then ALL FAIL
barrier ~ kT
Hard Sphere
Janus Colloids
Biphasic MixtureGel-Glass competition
Depletion Gels
SOFT “MOLECULAR”
Colloids and Nanoparticles : A Zoology of Material Systems
Stars
Microgels
* Large Effects of Particle Softness & Overlap
* Coupled Activated Translation-Rotation Dynamics * Glass vs. Gel vs. Attractive Glass vs. Plastic Glass vs. Double Glass
Alpha (Segmental) Relaxation Map: Thermal Liquids
MCTCrossover Region
High Temperature Arrhenius
)(TLog!
10!
7!
42~ !
T
1
cT
1
gT
1
!"" )(~
cTT
AT
1
Ae!"
Tc : “dynamic crossover” ~ 10-7 s
Below Tg NONequilibrium
Arrhenius…why ?
Aging….how ?
Effect of Stress ?
Tg!150"500 K
Polymers
local nanometer scale physics
DEEPLY Supercooled NON-ArrheniusT–dependent barrier
NLE Theory of Molten Segmental Dynamics
“Liquid of Gaussian Segments”
σ!!"jr
Displacement
!sdr(t)
dt= " d
drFeff(r(t)) + #(t)= 0
COARSE GRAIN to segment nm scale
! = C"l
S(q~0) ! S0
= " kBT #
T$ (%")2 "&2 $ &A+B
T
'()
*+,
&2
KEY : Dimensionless Compressibility or inverse Bulk Modulus ~ amplitude of nm scale density fluctuations
measurable via Scattering or Thermo …key to dynamical predictive power
r(t)
Local Angstrom Cage scale “chemistry” Averaged Over
T, P, N, chemistry dependentCohesive Energy Density key
Intra-Segment scale fast Arrhenius Segment-Segment forces
simplicity
emerges
S(q)“fast”
!(T ) " 1
#$3S03/2
SINGLE parameter quantifies Localizing Forces Feff (r)
λ > λc
λ < λc = 8.3
*
r /σincreases as cool or pressurize
Mean Segmental Hopping (alpha) Time
!"(T ) = !ARR
(T )eac#FB(T )
smooth crossover at Tc Arrhenius to Supercooled
σ!!"jr
* POLYMER Stiffness Effect : ac Cooperative Segment Hopping
DYNAMIC SEGMENT Kuhn or Persistence Length …MAP ac! " l
K=C
#l OR lp = C#+1( ) / 2$ ac ~1%9
NON-Arrenhius NON-analytic in T NON-singular at T > 0
a priori , sensible ~ 280 K, PDMS 460 K, PolyEtherImide
FB! "#"
c( )1.4
Barrier
Kramer’s
Inspired
Deeply Supercooled a-PMMA Melt
“Glass”
Expt Tg ! 378K if ac ! 5,6 ...predict m ~ 120!130 ~ expt
m !
d(log"#
)
d(Tg / T )Tg+
*
CH3CH2-C-
|
O=C-O-CH3|
Dynamic Fragility
~ range of MOST polymers
m ~ 55 --> 140
General Message : to leading order, fragility determined by backbone stiffness
Can FIT calculations VERY well to singular VFT or WLF …..but not true in NLE !
!" # exp
B
T$T0
%
&'
(
)*
m!16+40 a
c
•
1 new material parameter : b SAXS Expts : b ~ 0.4 - 0.7
SAXS: S(q ~ 0) = S0
S0(T < T
g) ! bS
0(T
g) +
T
Tg
(1" b)S0(T
g)
melt
glass
T < Tg:Collective Density Fluctuations in NONequilibrium GLASS
Basic Landscape Concepts
Frozen (δρ)2Inherent structures
Equilibrated ~ Vibrations
Tg
*
T
Teff
Tg
Alba-Simionesco et alMacromolecules, 2008
Confining FORCES:same physical picture as ABOVE Tg
Age
linear
“quenched”
Crossover to Arrhenius Relaxation (modestly) below Tg
b = 0 0.33 0.5 0.67 0.75
Activation Energy RATIO R ~ 3 - 6
PHYSICS : NONequilibrium effectEXPTS : diffusion,
dielectric, mechanical…. R = E
A
+/ E
A
!" 3 ! 6
disordered solid (!")2.... S
0(T )
Morefrozen (δρ)2
Arrhenius for all b Weaker Constraints
*
* ignore physical AGING….. “rapid quench experiment”
Equilibrium
Age of Universe !
(quenched) Glass Linear Elasticity
Consistent with Mechanical Definition of Glass via E’(T)
Young’s Modulus + Stress-Stress TCF + Green-Kubo +Maxwell
E ' ! 2.8
60!2dq0
"# q4 $lnS(q)
$q%
&''
(
)**
2exp +q2r
loc2 /3S(q)%
&'()* * r
loc
1.5 GPa @ Tg
~ linear
σ ~ (lp , lK ) ~ (0.75, 1.35) nm
a-PMMA
Mulliken & Boyce
Feff(r)
fit
MUCH stronger growth in liquid
Enthalpy, Density,… Modulus Yield Stress…..
Glass is NONequilibrium state…..…. Time-Dependent Properties more “solid-like” as equilibrate via local activated α -process
Sigmoidal Response McKenna, JPCM, 2003 equilibrium
!"! t
age
µ
exponent : µ(T) grows with cooling
plastics used “close” to TgQuiescent Physical Aging
Relaxation Time
Age Logarithmically ~ Log{tage}
“Down-Jump” Expt : quench below Tg and wait
ThermoMechanical Properties
Theory of Physical Aging
aging
S
0(t) ! " (#$)2 ! % F
B(t) & % '( (t), G ', '
y&
dS0
(t)
dt= !
S0
(t)! S0,l
"#
(t;S0
(t))
S0, l
: final equilibrium value
S0, g
: initial nonequilibrium value
PMMA
as time passes, equilibrate via Activated Hopping
PRL, 2007PRE,2008
NON-linear & Self-Consistent… No adjustable parameters
4% differ @ Tg-T = 8 K
Tg
“down jump” expt
ANSATZ
S0(t) = S
0 , l+ S
0(t = 0) ! S
0 , l( )exp ! dt ' "
#
!1
(t)
0
t
$%&'
()*
! logarithmic, slope " as cool
Density fluctuations, Cohesive energy, Modulus,…..
Aging Predictions for a-PMMA
log-log plotnormalized time, τα(0)
DOWN-JUMP :
Good Power laws
!" ! tage
µ
UP-JUMP :365 K --> 370 K VASTLY different: long incubation..then not power law
Correct Asymmetry
Tg = 378 K
S0,g
! S0
(t )
S0,g
! S0,l
Down Jump Power Law : T-dependent Aging Exponent
!" ! tage
µ(T )
Deeper Quench
Higher exponent
Faster AgingPS
NONuniversal Material Aspects
Struik “downturn”
Mechanical Behavior of Amorphous Polymer Plastics
1) Elastic solid
2) Strain Softening + “overshoot yield peak”
3) Dynamic Yielding segment scale “plastic flow”
Deformation-induced de-vitrification Tg(stress)
4) Strain Hardening uniquely polymeric…chain deformation
Physical Aging
Stress-Driven “Rejuvenation”…mechanical disordering…new steady state
coupled effects on mechanics
Stress - Strain
1
2
3
4
~ 5-10%
Fix T & Rate pre-aging time
* Flow Stress
* Simulations (Rottler, Robbins, dePablo…) * Experiments (Ediger,…)
Tight Correlation of isotropic SEGMENT Dynamics and Bulk Mechanical Properties
*Strong T & Strain Rate Dependence …..”Activated Process”
0th order Incorporation of Stress in NLE Theory
τ = Applied Stress
Reduces Modulus & Lowers Barrier
Accelerates Relaxation
External force on particle
r
F(r;!)=F(r;! =0)"##2! r
τ increases
!hop
!0
=2" g(# )
K0(! ) K
B(! )
eFB(! )
G '(! )= 1
60"2dq0
#$ q4
% lnS(q)%q
&
'(
)
*+
2
e,q2r
LOC2 (! )/3S(q)
ala isotropic Eyring @ “instantaneous dynamical variable” level
Mechanical Work
*
Stress “tilts landscape”
Kobelev+KSS PRE 2005
FB
(! ) " FB
(0) 1#(! /!y,abs
)$%&
'()
5/2
(transient) Glassy Modulus
rLOC
BELOW Hardening regime :little change in conformation, packing,…
Barrier Softening & “Stress-Induced De-vitrification”
Stress
Melting
…glass flows !
Input to Constitutive Eqn
1000 s
Barriera-PMMA GLASS
Tg = 378 K
T < Tg
* Neglect Aging “rapid quench”
*NEGLECT Mechanical Induced Structural Disorder
S0! f ( !" , stress, t) soley due to
“Landscape Tilting”
Generalized Maxwell Constitutive Eqn
0( '' )';( ) ( )
t
sE t t tresst d tt !" #= $ !
Ansatz : Nonlinear Boltzman superposition
“stretched” Maxwell Model:
E(t ! t ';stress) = E '(" (t '))exp ! dt ''"#
!1 " (t '')$% &'t '
t
()*
+,
! (t) = dt ' E '(! (t '))e
" dt ''!#"1 ! (t '')$
%&'t '
t
(!) (t ')
0
t
(
• Constitutive Equation
/ 2(1 ) 2.5 3E G !" " = + #!
“Effective Time” form
APPLY : constant strain rate
Nonlinear Self-Consistent Eqn for Stress(t)
Strain ! " = !" t
MacromoleculesJCP, 2008
(minimal) Input Physics : Stress-dependent Elastic Modulus & α−Relax Time
Comparison with a-PMMA Expts (rapid quench)
!! = 0.001
0.005
0.0001 sec"1
Tg-T = 97 K79674227
Hasan, Boyce et al. JPSPP, 1993
Lee & Swallowe, JMS,2006
Dynamic FLOW STRESS
τy(T) ~ linear in temperature
Yield STRAIN ~ 5-10%
Alpha Time AT Yield “short”….glass melting
FIT
100 s
FIT
yield
“Deformation Thinning” in Flowing GLASS per Complex Fluid
!! "# (0)( )c<<1
Master Curvemaster curve …per Ediger Expts & Sims
Riggleman, Schweizer, dePablo, Macromolecules 2008 ISOTROPIC acceleration TOO !
*
…“thinning” onset at
Plastic Flow regime “locally equilibrated” liquid-like nanodomains
!! "#,yield$%
&'<<1
More at higher T / Lower rates
! !" #0.9±0.1
Log(Pe)=
power law
a-PMMAMontes et al, Macromolecules, 2008
“Hardening Modulus” GR
Recent Experiments &Simulations (Hoy & Robbins; Lyulin,…)
• ala Ideal Entropic Intrachain Rubberλ−Dependence
**BUT**
• Magnitude > 100 times rubber Modulus
• Increases with Cooling…NOT entropic !
• Strain Rate dependent
• Correlates with Yield Stress
Suggests local dissipative process in glass relevant…SEGMENTAL RELAXATION
Classic Model qualitatively WRONG
Strain Hardening
λ
Theory : Suppressed Density Fluctuation Origin
Uniaxial Deformation --> Anisotropic single Chain Conformation
induce anisotropic INTERchain correlations….POST-Yield ”local equilibration”
PRISM THEORY : reduces density fluctuation amplitude, S0(λ) ….origin of deformed rubber network has INCREASED Tg
Enhances dynamical constraints, Barrier --> Slows Relaxation, Raises E’
λ
0 1 2 3 4 50.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.0, 501.0, 1001.0, 2500.5, 1002.0, 100
λ
�
S(!)S(1)
Anisotropic PRISMStrained rubber networks& melts Oyerokun & KSS, 2004
S0(!)
S0(1)
=1
1+3S
0(1)
16
"r
X
!2+2!#1
3#1
$
%&
'
()
*
large stress upswing…..”Strain Hardening” due to Glassy Physics
Length Scale of Affine Deformation ? (effective crosslinks) scale beyond which glass = “solid”
rX ~ 4 nm ~ Scattering & Mechanical PMMA Expts , Tg-T ~ 20 K….NX ~ 16 Montes et al, Macromolecules, 2008
PRL, 2009
NONmonotonic Alpha Relaxation Time
Enhancement MODEST (Ediger Experiments)
…..BUT “big enough”
Tension
Compression
CoolIncrease Strain Rate
Vary Tg-T = 90, 70, 50, 30, 10 @ rate = 0.001 s-1
Vary RATE = 10-5 - 0.1 sec-1
@ fixed Tg-T = 50
NX=16
Extract Hardening Modulus g(!) = !
2" !
"1
Hardening
Correct “rubber-like” form
Dynamic Yield Stress
!y(T , !" )
GR
Yield
Many Open Issues
GR ~ 100 Ge ~ even close to Tg
~ Linear with T
~ Log{strain rate}
correlates with Yield Stress
absolute numbers reasonable (factor 2-4)
Tension
Yield Stress@ 10-3 s-1
Theory Results : NON-Entropic Hardening Modulus
Predict non-affine length scale ?….rate,T dependence ? micro- vs. macro- stretch ?
role of entanglements, Ne ? role of chain scale dynamics ?
tension vs compression asymmetry ?
Compress
Rob HoyMark RobbinsCorey O’HernRob Riggleman
* NEED: Exptl Probe Density Fluctuations in Hardening Regime….SANS, USAXS ?
strain rate 10-1 Hz 10-3
10-5
Other Phenomena & NLE Polymer Glass Theory
Creep ; Creep RecoveryStrain Softening, Yield Peak
due to COUPLED Mechanical Rejuvenation & Physical (pre) Aging JCP, 2008
PRE, 2010+ in progress
Lee, Ediger, et al, Science, 2009
Nature of Nonequilibrium Steady State ? aging erasure…..Viscosity Bifurcation: aging under stress Aging wins vs. Rejevenation wins ?
S0= f ( !! , stress, tpre"age,t)
MANY Open Questions & Theoretical Challenges
Supercooled Melts : * Decoupling of Segment Scale & Chain scale relaxation
recent new idea : Alexei Sokolov & KSS, PRL, 2009
* Dynamic Heterogeneity effects, KWW relaxation, Chain modes….
Polymer Glass : * Kovacs aging “memory effect”…multiple temperature jumps
* Tensorial deformations : tension vs. compression vs. shear…..
* Role of Dynamic Heterogeniety in “solid state” mechanics ?
Polymer Nanocomposites : marriage of the colloidal and polymeric worlds !glasses and gels relevant
Colloidal Glasses & Gels : nature of Aging, Mechanical Rejuvenation,…..
Beyond NLE : 2-particle dynamic correlations Daniel Sussman & KSS, JCP, submitted, November, 2010